Stay Or Switch?

[DFCarlson]

The Problem:

You are a guest on a game show, and you are confronted by three closed doors. The spaces behind two of the doors are empty, and the space behind one of the doors contains $1,000,000 in cash. You are asked to choose one of the three doors and if the $1,000,000 is behind the door you choose you get to keep the cash.

You choose a door.

Before the door you chose is opened, the game show host, who knows where the $1,000,000 is located, opens one of the remaining doors. The space behind the door that the host opens is empty. You are then given the choice to stay with your original choice or switch to the second of the remaining doors.

Is it to your advantage to stay with the door that you originally chose or should you switch your choice to the remaining door? Or doesn't it make any difference? Why?

Note: This is the original statement of a problem known as the Monty Hall dilemma and it has been widely discussed on the internet and in several books, but there are still those who disagree about what the correct answer is.

[Taijitu]

Check this out.

Monty Hall Dilemma

[DFCarlson]

Check this out.

Monty Hall Dilemma

Do you agree with the answer here? Or do you see other possibilities?

Others have said that after the host opens a door and the room is empty, you are left with two unopened doors. You can disregard the opened door, it is now out of the game. Therefore your chance that the $1,000,000 is behind one of the two doors is 1 in 2 and it doesn't matter whether or not you switch.

What do you think?

Edited July 28, 2007 by DFCarlson

[Pink Mist]

you have a one in three chance to start and even with the open door, it's still one in three

[DFCarlson]

you have a one in three chance to start and even with the open door, it's still one in three

You are right, at first you have a 1 in 3 chance when there are 3 unopened doors, but when the host opens one door the situation changes. You are now given the opportunity to choose between 2 doors, the one you originally chose and the one remaining from the other 2. That seems to indicate that you now have a 1 in 2 chance to pick the correct door since the situation has changed from what it was originally when all 3 doors were closed.

[canadiangirl]

The Problem:

You are a guest on a game show, and you are confronted by three closed doors. The spaces behind two of the doors are empty, and the space behind one of the doors contains $1,000,000 in cash. You are asked to choose one of the three doors and if the $1,000,000 is behind the door you choose you get to keep the cash.

You choose a door.

Before the door you chose is opened, the game show host, who knows where the $1,000,000 is located, opens one of the remaining doors. The space behind the door that the host opens is empty. You are then given the choice to stay with your original choice or switch to the second of the remaining doors.

Is it to your advantage to stay with the door that you originally chose or should you switch your choice to the remaining door? Or doesn't it make any difference? Why?

You want to switch to maximize your chances. But it's certainly not immediately intuitive why.

What makes the difference is that the host knows which door has the money. When he choose a door to open, he chooses a door with no money behind it. He also doesn't choose "your" door.

So let's look at all the possibilities:

Let's say the money is behind door #1:

1. You pick door #1 first, and then you switch -- LOST

2. You pick door #1 first, and then you stay -- WIN

3. You pick door #2 first, and then you switch. The host will have opened door #3, so you will pick #1 -- WIN

4. You pick door #2 first, and stay -- LOST

5. You pick door #3, and switch. Since the host can't pick #1 (it has the money), and #3 (you've picked it), he'll pick #2, so you'll pick #1 -- WIN

6. You pick door #3 and stay -- LOST

Out of the three situations where you decide to switch, you win 2/3 times

Out of the three situations where you stay, you only win 1/3 times.

So the best strategy is to switch.

[DFCarlson]

The Problem:

You are a guest on a game show, and you are confronted by three closed doors. The spaces behind two of the doors are empty, and the space behind one of the doors contains $1,000,000 in cash. You are asked to choose one of the three doors and if the $1,000,000 is behind the door you choose you get to keep the cash.

You choose a door.

Before the door you chose is opened, the game show host, who knows where the $1,000,000 is located, opens one of the remaining doors. The space behind the door that the host opens is empty. You are then given the choice to stay with your original choice or switch to the second of the remaining doors.

Is it to your advantage to stay with the door that you originally chose or should you switch your choice to the remaining door? Or doesn't it make any difference? Why?

You want to switch to maximize your chances. But it's certainly not immediately intuitive why.

What makes the difference is that the host knows which door has the money. When he choose a door to open, he chooses a door with no money behind it. He also doesn't choose "your" door.

So let's look at all the possibilities:

Let's say the money is behind door #1:

1. You pick door #1 first, and then you switch -- LOST

2. You pick door #1 first, and then you stay -- WIN

3. You pick door #2 first, and then you switch. The host will have opened door #3, so you will pick #1 -- WIN

4. You pick door #2 first, and stay -- LOST

5. You pick door #3, and switch. Since the host can't pick #1 (it has the money), and #3 (you've picked it), he'll pick #2, so you'll pick #1 -- WIN

6. You pick door #3 and stay -- LOST

Out of the three situations where you decide to switch, you win 2/3 times

Out of the three situations where you stay, you only win 1/3 times.

So the best strategy is to switch.

Yes, you're right - I've been playing Devil's advocate...

Another way to put it is that when you pick one door, the probability that you've won is 1 in 3 and the probability that the money is behind one of the other doors is 2 in 3.

When the host, who knows where the money is, opens one of the remaining two doors, that doesn't change the fact that the probability that the money is behind one of the two doors that you didn't pick is 2 in 3 - but now there's only one door left of the two that you didn't pick.

So the chance that the money is behind the door you did pick is still 1 in 3 and the chance that the money is behind the remaining door that you didn't pick is 2 in 3. So you are better off switching.

Now.. suppose that the game show host is ignorant of which door it is that the money is behind. In this case, should you switch or not?

Edited July 28, 2007 by DFCarlson

[slapout]

Why is this called the Monte Hall delemma? This was the old nut under the cup or shell game used through out gambling history. Common to show one empty shell to induce losers to stay a while longer, lets get every coin they have if possible.

[DFCarlson]

Why is this called the Monte Hall delemma? This was the old nut under the cup or shell game used through out gambling history. Common to show one empty shell to induce losers to stay a while longer, lets get every coin they have if possible.

Monty Hall was the name of the game show host on an old television show which used the same method as described in the problem. (BTW it's not the same as the old shell game.)

[ratchabuild]

Why is this called the Monte Hall delemma? This was the old nut under the cup or shell game used through out gambling history. Common to show one empty shell to induce losers to stay a while longer, lets get every coin they have if possible.

Monty Hall was the name of the game show host on an old television show which used the same method as described in the problem. (BTW it's not the same as the old shell game.)

It's evens or levels you devils.

[Wizzard of Oz]

Check this out.

Monty Hall Dilemma

Wow, this is cool. I was gonna go with NO WAY MAAAAAN!

But you can't argue with a truth table.

you have a one in three chance to start and even with the open door, it's still one in three

Lol. When someone gives you an onion and tells you it's an apple and you start eating it and crying from the onion's vapours, you will adamantly say that it is an apple right?

Edited July 29, 2007 by Wizzard of Oz

[Pink Mist]

lies, lies and statistics, who do you beleive, lawyers, accountants or statiticians?

[jasreeve17]

I see the problem as two seperate ones.

The first stage is always a one in three chance.

The second stage is always a one in two chance.

Connecting the two makes presumtions which I don't think are as clear, or correct, as stated by the linked site.

I understand the law of probability, but life's experiences tell us that Murphy's law is more powerful. I wouldn't switch doors, but that may be why I'm poor...

[Pink Mist]

how can switching make your probable chances increase, the prize hasn't shifted doors, how can the probability?

[jasreeve17]

how can switching make your probable chances increase, the prize hasn't shifted doors, how can the probability?

Because they are linking it to the fact that the guy opens a door with knowledge of what's behind it. In effect adding another instance that acts upon the original dilema, and thus changes it.

I don't agree with that though.

[Pink Mist]

how can switching make your probable chances increase, the prize hasn't shifted doors, how can the probability?

Because they are linking it to the fact that the guy opens a door with knowledge of what's behind it. In effect adding another instance that acts upon the original dilema, and thus changes it.

I don't agree with that though.

but that isn't the paradox, thats an assumption

[jasreeve17]

how can switching make your probable chances increase, the prize hasn't shifted doors, how can the probability?

Because they are linking it to the fact that the guy opens a door with knowledge of what's behind it. In effect adding another instance that acts upon the original dilema, and thus changes it.

I don't agree with that though.

but that isn't the paradox, thats an assumption

I agree with you which is why I don't agree with the paradox.

(I'm not convinced by the assumptions.)

[Soju]

I can't believe that nobody came up with the correct solution to this puzzle. The correct answer is that it is impossible to determine. The reason being, the original statement is ambiguous, which was also the case when this puzzle was first introduced many years ago.

Based on your replies, it can be assumed that what you really meant was:

The host, who knows where the money is, always opens one of the two remaining doors to reveal nothing.

If you word the statement that way, then the odds are in your favor (2 out of 3) to switch.

The fact that the host knows where the money is, is totally irrelevant to this puzzle as stated because there is absolutely no mention of what he does with that information. If he chooses to ignore it, or to use that information against you, or to use that information to help you as much as possible, then the odds have totally changed. Some examples, depending on what the host does with that information:

The host knows where the money is but opens one of the remaining doors only if you have choosen the money.

In this case you will lose 100% of the time if you switch

The host knows where the money is but only opens an empty door if you have not chosen the money.

In this case you will win 100% of the time if you switch

The host knows where the money is, but ignores that fact and just opens randomly one of the two remaining doors.

In this case, 1/3 of the time the host will open the door with the money, and you will lose, assuming you cannot choose a door once the money has been revealed. In the other 2/3 cases, half the time you will switch to win the money, and half the time switch away from the money. So the end result is that you win in only 1/3 of the total cases, but 1/2 of the cases if an empty door is revealed.

The host knows where the money is, but does not open the two remaining doors based on any systematic approach.

In this case, the odds of winning by switching are impossible to determine.

Your question,

Now.. suppose that the game show host is ignorant of which door it is that the money is behind. In this case, should you switch or not?

would be identical to the case where the host knows but opens a random door. In that case, once a door is opened and if it reveals nothing your odds are 50:50 by switching, so nothing gained or lost on average by doing the switching. But as already mentioned, your overall winnings will only be 1/3 being that 1/3 of the time the host will reveal the money, disallowing you to then choose that door. This is probably the scenario most people will think of when trying to solve this problem, so answer accordingly, without realizing just how important the game show host's actions with respect to what knowledge he has are, and how drastically it can affect the solution.

Anyways, it seems that most puzzles are stated in an ambiguous manner because what seems to be unambiguous to the author is found to be ambiguous by the readers. And sometimes the only way to be unambiguous will make the puzzle's answer be too obvious. In many cases though the intention of the puzzle is easily determined, so the ambiguity doesn't create any problems. But in this case, I'd say the there is no clear-cut way to determine what the real intention of the puzzle is, as stated, unless you see the answer. This is because the case where the game show host intentionally tries to make you lose , at least part of the time, can be assumed by many to be a very likely scenario. A very good puzzle, that although very easy to solve makes you think carefully first.

Edited July 30, 2007 by Soju